This preprint is a terminal certificate-based sequel to the author's TEBAC Birch--Swinnerton-Dyer master manuscript over \ (Q\). It consolidates the post-master modules developed in the TEBAC BSD program into one submission-facing proof dossier: \ (VII-D3--VII-D7 + VIII-A + VIII-B + VIII-C + IX\). The consolidated chain consists of the analytic determinant gate, the central Selmer--Kummer bridge, the integral lattice and regulator handoff, the residual Selmer/Cassels--Tate/Sha/Tamagawa/torsion finite ledger, and the final leading-coefficient assembly. The main result is the non-circular terminal implication \ (Z₁ₒ₃ (E) =0 BSD (E/ Q) \), where \ (Z₁ₒ₃ (E) = (Z₀₍, Zₒ₊, Z₋₀ₓ, Z₅₈₍, Z₄ₑ, Z₋₄₀₃, Z₅₎ₑ₁) \). Thus, if the printed analytic, Selmer--Kummer, lattice/regulator, finite-ledger, period, leading-coefficient, and forbidden-import defect rows vanish, the BSD rank formula and leading-coefficient formula follow for elliptic curves over \ (Q\): \ (ordₒ=₁L (E, s) =rankE (Q) \), and, for \ (r=rankE (Q) \), ^ (r) (E, 1) r!=EReg (E) | (E) |ₚ cₚ (E) |E (Q) ₓ₎ₑₒ|². \ The paper is deliberately claim-safe. It does not assert that the Birch--Swinnerton-Dyer conjecture is unconditionally resolved independently of the displayed terminal certificates. Instead, it presents a finite certificate system and a precise dependency graph. The remaining constructor-level tasks needed for a fully unconditional proof are explicitly separated: actual primitive trace state-space construction, actual prime-packet construction, endpoint estimates, residual Schur gap from the trace model, actual Selmer--Kummer realization, and finite Sha/Cassels--Tate closure. This record is a terminal certificate-based sequel to the author's earlier TEBAC BSD master manuscript and its associated Core modules. The earlier records are cited below by DOI as related identifiers.
Tosho Lazarov Karadzhov (Sun,) studied this question.